AP Calculus AB and BC

Unit 7: Differential Equations

Exam weighting: AB 5-10% · BC 5-10%

Unit 7 covers differential equations, which relate a function to its derivatives. You model situations, verify solutions, read slope fields, and use separation of variables to find general and particular solutions, including exponential and logistic growth. It is 6-12% of the AB exam and 6-9% of the BC exam.

The core idea

A differential equation gives you a function's rate of change and asks you to recover the function itself. Where earlier units differentiate a known function, Unit 7 runs the process backward: you are handed dydx\frac{dy}{dx} in terms of xx and yy, and your job is to find every yy that fits. The whole unit is one recognition question asked in different forms: given this differential equation, do I verify a solution, describe it qualitatively with a slope field, or solve it outright by separation of variables?

The unit moves from setup to solution. Topic 7.1 translates a sentence into a differential equation, and Topic 7.2 checks a candidate solution by substituting it back. Topics 7.3 and 7.4 read solutions qualitatively from a slope field, a grid of short segments whose slopes are dydx\frac{dy}{dx} at each point, and BC adds Euler's method (Topic 7.5) to step along a solution numerically. Topics 7.6 and 7.7 solve equations exactly by separation of variables, Topic 7.8 applies that machinery to exponential growth, and Topic 7.9 (BC only) interprets logistic growth models qualitatively, reading carrying capacity and fastest-growth point without solving the equation.

  • Verify a solution (Topic 7.2): the problem hands you both an equation and a candidate function. Differentiate the candidate, substitute into the equation, and check that both sides match. You never solve, you confirm.
  • Slope field (Topics 7.3 and 7.4): the problem shows a grid of segments or asks you to sketch one. Plug points into dydx\frac{dy}{dx} to get slopes, then trace a solution curve by following the segments. No algebra required.
  • Separation of variables (Topics 7.6 and 7.7): the problem says solve or find the particular solution, and dydx\frac{dy}{dx} factors into a function of xx times a function of yy. Separate the variables, integrate both sides, then apply the initial condition.
dydx=g(x)h(y)  ⟹  ∫1h(y) dy=∫g(x) dx\frac{dy}{dx} = g(x)h(y) \;\Longrightarrow\; \int \frac{1}{h(y)}\,dy = \int g(x)\,dx

Integrating both sides introduces a constant +C+C, so separation of variables produces a general solution: a whole family of curves, one for each value of CC. A single initial condition, such as y(0)=2y(0) = 2, pins down one particular solution. Solve for CC using the initial condition right after you integrate, before simplifying, and put the +C+C on only one side. Watch for domain restrictions: a particular solution is valid only on the interval around the initial point where it stays defined.

Two word-problem patterns collapse into standard differential equations. When a rate is proportional to the current amount, the model is exponential (Topic 7.8): growth for k>0k > 0, decay for k<0k < 0. When the rate is jointly proportional to the amount and the room left before a carrying capacity, the model is logistic (Topic 7.9, BC only). For logistic growth you can read the carrying capacity aa as the value yy approaches as t→∞t \to \infty, and the population grows fastest at half the carrying capacity, both without solving the equation.

dydt=ky  ⟹  y=y0ekt\frac{dy}{dt} = ky \;\Longrightarrow\; y = y_0 e^{kt}
dydt=ky(a−y)\frac{dy}{dt} = ky(a - y)

What the exam asks

Unit 7 is 6-12% of the AB exam and 6-9% of the BC exam. Slope fields and separation of variables are staple free-response tasks, often paired: sketch the slope field, then solve the equation and match the two. Multiple-choice questions test verifying a solution (Topic 7.2) and setting up the right model (Topic 7.1). BC students additionally see Euler's method (Topic 7.5) and logistic models (Topic 7.9). Progress Check 7 has about 15 multiple-choice questions on AB and 20 on BC, plus 3 free-response questions.

Topics in this unit

Topic numbers and titles from the College Board Course and Exam Description.

  • 7.1Modeling Situations with Differential Equations
  • 7.2Verifying Solutions for Differential Equations
  • 7.3Sketching Slope Fields
  • 7.4Reasoning Using Slope Fields
  • 7.5Approximating Solutions Using Euler's MethodBC only
  • 7.6Finding General Solutions Using Separation of Variables
  • 7.7Finding Particular Solutions Using Initial Conditions and Separation of Variables
  • 7.8Exponential Models with Differential Equations
  • 7.9Logistic Models with Differential EquationsBC only

How to study this unit

  • Topics 7.2 and 7.6: read the verb. Verify a solution means substitute the given function and check that both sides agree; solve or find the general solution means separate variables and integrate. Doing the wrong one burns time on a task that wanted the other.
  • Topics 7.6 and 7.7: put all $y$ terms with $dy$ on one side and all $x$ terms with $dx$ on the other before integrating, write the $+C$ on the $x$ side only, and solve for $C$ from the initial condition immediately after integrating, not at the end.
  • Topics 7.3 and 7.4: to sketch a slope field, plug each grid point into $\frac{dy}{dx}$ and draw a short segment with that slope. When $\frac{dy}{dx}$ depends only on $x$ the segments match along vertical lines; when it depends only on $y$ they match along horizontal lines.
  • Topic 7.8: memorize that a rate proportional to the amount, $\frac{dy}{dt} = ky$, always solves to $y = y_0 e^{kt}$. Spotting that phrasing lets you write the answer without re-deriving it, with $k > 0$ for growth and $k < 0$ for decay.
  • BC only: for Euler's method (Topic 7.5) build a table and repeat $y_{i+1} = y_i + \frac{dy}{dx}\cdot \Delta x$ one step at a time; for a logistic model (Topic 7.9) read the carrying capacity as the limiting value and recall growth is fastest at half of it, both without solving.

Guides for this unit